Remark 1.1 The prefactor \(W(X)\)  is the same \(W(X)\)  in [1]} in the homogeneous case. It corresponds to the fact that the model is PT symmetric with respect to the invariant measure, and \(W(X)\) , like the \(C_q(\vec{n})\)  in [2]}, gives the invariant measure for the \(q\) -TAZRP process, up to a constant. See [2]} and [4]}.

